The five-dimensional Watson integral
Find a closed form for the five-dimensional Watson integral W5 ≈ 0.2312616250, which fixes the return probability of a random walk on Z^5.
Problems
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Loaded dice with a uniformly distributed sum
Show that if a sum of two independent integer random variables is uniform on its support, then each summand is uniform on its own support.
Openper Ben Green's 100 open problems, 2026-10-20
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The mean of the Tracy–Widom distribution
Find a closed form for the mean μ ≈ −1.7710868074 of the Tracy–Widom distribution of the largest GUE eigenvalue.
The site percolation threshold of the square lattice
Find a closed form for the site percolation threshold of the square lattice, p_c ≈ 0.5927460508; bond percolation has p_c = 1/2.
Maximal flow through oriented percolation as p tends to 1
Prove that the deficit of maximal flow through oriented bond percolation behaves like sqrt(2(1 − p)) as p tends to 1.
Openper David Aldous's open problems, 2026-10-20
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Online minimum spanning trees with random edge weights
Prove that the optimal expected cost of building a spanning tree online, with random edge weights, converges, and find the limit.
Openper David Aldous's open problems, 2026-10-20
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Are the largest roots of a random polynomial real?
Find the limiting probability that the largest roots of a random polynomial with Gaussian coefficients are all real.
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The Häggström–Pemantle conjecture on competing growth
Prove that two competing growth processes on the lattice cannot both grow forever when their speeds differ.
Openper AIM Problem Lists, 2026-10-20
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Empires and percolation: regions merging at a constant rate
Decide whether planar regions merging with neighbours at a constant rate form an infinite region in finite time.
Openper David Aldous's open problems, 2026-10-20
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The Chvátal–Sankoff constant for binary strings
Pin down the constant γ in the expected longest common subsequence γn of two random binary strings, rigorously between 0.79266 and 0.82628.
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The travelling salesman constant (Beardwood–Halton–Hammersley)
Narrow the constant β in the length β√n of the shortest tour through n random points, rigorously between 0.6277 and 0.8810.
Openper Optimization Constants in Mathematics, 2026-10-20
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